A LEMP stack is a group of open source software that is used to get webservers running. LEMP is different from your traditional LAMP (Linux Apache2, PHP, MySQL) stack as it uses NGINX (pronounced Engine X) and PHP-FPM which are both lightweight and optimized for performance without a lot of fine-tuning.
There is a lot of information out there on configuring WordPress with a lemp stack. Please note that not all information out there is good, and there is a lot of inherent security risks attributed with hosting your own websites. WP Jedi has a really good article on WordPress & Nginx Hosting Resources for those who want to learn more. There is also a really good repository on GitHub for Nginx WordPress Configurations.
Please be aware that this tutorial is for advanced users only. You must have a pretty good understanding of how Linux works as well as familiarity with general web server concepts. If you are looking for a simple way to host your website, please visit Who Is Hosting This for reviews of top web hosts.

Getting Started

There are a few requirements to take care of before beginning this tutorial. First, you must have a server that is running Ubuntu. I have tested this on the 12.04 LTS, 12.10, and 13.04 releases of Ubuntu, which are the three most recent releases. Next you must have ports 22 (for SSH/SFTP) and 80 (for web access) opened and SSH configured.

Step 1: Update

This tutorial will be using Ubuntu’s package manager apt-get to install all of your software. Running updates will allow you to use the latest version of the software you download, as well as any security patches. For any tasks requiring elevated privileges you must either prefix these commands with ‘sudo’ or run them as root. Run these commands:

# apt-get update
# apt-get dist-upgrade

Step 2: MySQL

Now that your operating system is up to date, it is time to install the server components. Let’s start with the database server, MySQL.

# apt-get install mysql-server

You’ll need to enter a root password. There aren’t any requirements, but I recommend making it as secure as you can.

Step 3: Nginx

Now that MySQL is set up, it’s time to install your web server to serve static files.

Conclusion

Now if you go to the IP address of your server from any web browser you will now see a working version of WordPress! Now that you truly have complete control over your website, the possibilities are endless.